• Screening
  • Pros: early detection

• Immediate treatment?
  • Case: erectile dysfunction
  • Cons: overtreatment

https://howardwolinsky.substack.com/p/the-big-lie-and-low-risk-prostate?s=r

• The blood level of PSA is often elevated in men with prostate cancer.

1. How can we make biopsy schedules in AS on an individual level?

2. How can our personalized schedules relieve the burden of biopsies in AS at acceptable cost?

• Measurement error

Tomer, A., Nieboer, D., Roobol, M. J., Steyerberg, E. W., & Rizopoulos, D. (2022). Shared decision making of burdensome surveillance tests using personalized schedules and their burden and benefit. Statistics in medicine, 10.1002/sim.9347.

• Target population: Patients diagnosed as Prostate cancer with Gleason score of 3+3

• Sample size: 850 subjects recorded in the dataset, among which 833 participants had adequate PSA measurements for analysis.

• Data types:

  • Repeated measurements: PSA values over time

  • Time-to-event data: time until progression

  • baseline variables: age at diagnosis and PSA density

• Competing risk: treatment before progression (87/833)

• Interval censoring: periodical examinations

• Longitudinal submodel:

• Longitudinal submodel:

\[ \begin{align} \log_2(\text{PSA}_i + 1)(t) &= \color{blue}{m_i(t)} + \epsilon_i(t)\\ \color{blue}{m_i(t)} &= \beta_0 + b_0 + \underbrace{(\beta_1 + b_1)\mathcal{C}^{(1)}_i(t) + (\beta_2 + b_2)\mathcal{C}^{(2)}_i(t) + (\beta_3 + b_3)\mathcal{C}^{(3)}_i(t)}_{\text{natural cubic splines for time variable}} + \underbrace{\beta_4(\text{Age}_i - 62)}_{\text{baseline covariate}} \end{align} \] where \(\mathcal{C}(t)\)is the design matrix for the natural cubic splines for time \(t\); the error term \(\epsilon_i(t) \sim t_3\)

• Survival submodel:

The hazard, \(h_i^{(k)}(t)\), of experiencing event \(k\) (\(k = 1\) indicates progression; \(k = 2\) indicates treatment) for subject \(i\) at time \(t\) is: \[ \begin{align*} h_i^{(k)}(t \mid \boldsymbol{\mathcal{M}}_i(t)) = h_0^{(k)}(t)\exp\Big\{\gamma_k\text{density}_i + \alpha_{1k} \color{blue}{m_i(t)} + \alpha_{2k}\big[\color{blue}{m_i(t)} - \color{blue}{m_i(t-1)}\big]\Big\} \end{align*} \] where:

• \(\boldsymbol{\mathcal{M}}_i(t)\) is the estimated true history of PSA levels for subject \(i\)

• \(\color{blue}{m_i(t)}\) is the estimate of the true PSA level for subject \(i\) at \(t\)

• \(\color{blue}{m_i(t)} - \color{blue}{m_i(t-1)}\) is the estimate of the PSA level change in the previous year from \(t\) for subject \(i\)

• Visualization of \(\color{blue}{m_i(t)}\)

• Visualization of \(\color{blue}{m_i(t)} - \color{blue}{m_i(t-1)}\)

More interest goes to the probability of getting progression, which can be achieved by cause-specific cumulative incidence function

\[\text{CIF}_j^{k=1}(t) = \int^t_0 \underbrace{h_j^{(1)}(s)}_{\text{instantaneous risk}\\ \text{of progression}}\underbrace{\exp\left\{-\sum^K\int^s_0h_j^{(k)}(\nu)d\nu\right\}}_{\text{overall survival function}}\]

• Two indicators

  • Number of biopsies to be conducted

  • Detection delay

• Higher threshold \(\rightarrow\) fewer biopsies but longer detection delay

• Lower threshold \(\rightarrow\) more biopsies but shorter detection delay

• 100 Training sets, 300 subjects
• 100 Test sets, 200 subjects
  • 100 with progression
  • 100 without progression
• Fit the joint models on the training set and create personalized schedules for the test set
• Compared with the fixed schedules based on:
  • Number of biopsies
  • Detection delay
• Fixed schedules include:
  • Annual biopsies
  • PASS protocol: at year 0.5, 1, 2 and afterwards biennially

• Impact of detection delay on the patients?

  • link with MISCAN

• Progression-specific risk

• Assume precision of biopsies

• The horizon of interest: \(t_{\text{hor}}\)
• Generated schedules: \(S_i^\kappa = \{s_1, ..., s_{N_i}\}\)
• Progression is not observed when generating the schedules
• Expected number of biopsies are calculated
  • List all the possibilities of occurrence of progression

\[ \begin{align*} \mathit{Nb}_i(S_i^{\kappa}) = \begin{cases} 1, &t < T^*_i \leq s_1, k=1 \\ 2, &s_1 < T^*_j \leq s_2, k =1 \\ ... \\ N_i, &s_{N_i - 1} < T^*_i \leq s_{N_i}, k=1 \end{cases} \end{align*} \]

Generally, the expected number of biopsies:

\[ \begin{align*} E\{\mathit{Nb}_i(S_i^\kappa)\} = \sum_{n=1}^{N_i} n \times \underbrace{\Pr\{s_{n-1}<T^*_i \leq s_n | T^*_i \leq s_{N_i}, k =1 \}}_{\text{probability of progressing between two biopsies}}, \ \ s_0 = t \end{align*} \]

List all the possibilities of occurrence of progression:

\[ \begin{align*} \mathit{Dd}_j(S_j^{\kappa}) = \begin{cases} s_1 - T^*_i, &t < T^*_i \leq s_1, k=1 \\ s_2 - T^*_i, &s_1 < T^*_i \leq s_2, k =1 \\ ... \\ s_{N_i} - T^*_i, &s_{N_i - 1} < T^*_i \leq s_{N_i}, k =1 \end{cases} \end{align*} \]

Generally, the expected detection delay is:

\[ \begin{align*} E\{\mathit{Dd}_j(S_i^\kappa)\} = \sum_{n=1}^{N_i} \big[s_n - \underbrace{E(T^*_i | s_{n-1}, s_{n}, v, k=1)}_{\text{expected progression time}}\big] \times \underbrace{\Pr\{s_{n-1}<T^*_i \leq s_n | T^*_i \leq s_{N_i}, k =1 \}}_{\text{probability of progressing between two biopsies}} \end{align*} \]